RainySeptember, I don't think you have much insight into how grief can affect people and cause ructions in a family dynamic.
The Sister here comes across as the person who is determined to look like the Golden Child, the martyr, she who sacrifices all.
The OP faithfully reports all the shit she has been dished up about the brother, including the little detail that the grass doesn't look cut. A little bit of thought and maybe a little experience of lawncare should have caused the OP to get suspicious at that point - if the grass gets cut, let's say, every weekend that the sister is away, then of course it will have a week's growth by the time the sister sees it.
Then we have the details of the sister living in a one room flat, meanwhile the brother and his family were 'between houses ' at one stage and living with the DM, and then the detail about the brother and his wife buying a holiday home, and maybe we get a glimpse here of the green eyed monster? I suspect someone is feeling quite sorry for herself.
And the request that the brother take the DM for a few weeks of holiday or cut short the holiday to be with the mother is presumptuous and based on the sister's own self imposed predicament.
Nobody knows how the brother's wife might feel about any of this and apparently the sister did not ask her or include her in the narrative of what a wrong 'un the brother is. Maybe the wife has family of her own whom she has neglected during the I'll less of the DF, and has plans for the six weeks that involve them? Maybe the brother and his wife shouldered a lot of the work involved in caring for the DM during the DF's illness? The sister lives too far away to have been much practical use at that time, apparently.
What the OP has reported is incredibly one sided, and comes across as a carefully constructed piece of mud slinging.
Grief can also cause depression and can exacerbate anxiety, and some people use anger about the behaviour of others as a means of getting through bereavement. Not consciously - it can all seems perfectly reasonable and warranted, and perfectly coherent narratives involving caring daughter and feckless son can be constructed and spread around.
Depression or anxiety can also lead to a focus on some other person - anxiety about the mother, anger about the brother. It's not necessarily as critical as the sister believes it is that the DM has family for company at this point.
If I were the OP, I would take it upon myself to take a little care of the sister at this time.